## Matrix AlgebraMatrix Algebra is the first volume of the Econometric Exercises Series. It contains exercises relating to course material in matric algebra that students are exoected to know while enrolled in an (advanced) undergraduate or a postgraduate course in econometrics or statistics. The book contains a comprehensive collection of exercises, all with full answers. But the book is not just a collection of exercises; in fact, it is a textbook, though one that is organized in a completely different manner than the usual textbook. The volume can be used either as a self-contained course in matrix algebra or as a supplementary text. |

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### Contents

IV | 1 |

V | 4 |

VI | 11 |

VII | 15 |

VIII | 19 |

IX | 39 |

X | 43 |

XI | 47 |

XLIV | 265 |

XLV | 273 |

XLVI | 274 |

XLVII | 281 |

XLVIII | 284 |

XLIX | 292 |

L | 295 |

LI | 299 |

XII | 61 |

XIII | 67 |

XIV | 73 |

XV | 75 |

XVI | 83 |

XVII | 87 |

XVIII | 97 |

XIX | 98 |

XX | 103 |

XXI | 109 |

XXII | 119 |

XXIII | 126 |

XXIV | 131 |

XXV | 132 |

XXVI | 137 |

XXVII | 143 |

XXVIII | 148 |

XXIX | 151 |

XXX | 155 |

XXXI | 158 |

XXXII | 175 |

XXXIII | 182 |

XXXIV | 187 |

XXXV | 192 |

XXXVI | 201 |

XXXVII | 209 |

XXXVIII | 211 |

XXXIX | 228 |

XL | 231 |

XLI | 243 |

XLII | 246 |

XLIII | 255 |

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### Common terms and phrases

block commute complex numbers Consider decomposition defined denote determinant diagonal elements diagonal matrix differential duplication matrix dvec Econometric eigenvalues eigenvectors eigenvectors associated elementary Equality occurs equation example exists exp(A follows from Exercise full column rank Hence Hermitian Hessian matrix idempotent matrices implies inequality inner product inner-product space inverse Jordan chain Kronecker product linear combination linearly independent m x n matrix function matrix of order MP-inverse multiplication n x n matrix nonnegative nonsingular matrix nonzero eigenvalues null vector obtain orthogonal matrix permutation polynomial positive definite positive semidefinite positive semidefinite matrix Premultiplying proof prove real numbers result follows rk(A rk(B row rank satisfying Schur's Section Show Solution Let span square matrix submatrix subspace symmetric idempotent symmetric matrix theorem transpose triangular matrix unique unitary upper triangular vech(A vector space write x'Ax zero